thermodynamica 2 thermodynamic relations of systems in ... · thermodynamica 2 thermodynamic...

November 23, 2010

1

Thermodynamica 2 Thermodynamic relations of systems in equilibrium

Thijs J.H. Vlugt

Engineering ThermodynamicsProcess and Energy Department

Lecture 5

November 23, 2010 2

Today:• Equation of state 11.1• Departure functies 11.7

• Homework: Moran&Shapiro 11.68, 11.69, 11.70 • Homework: extra opgaven 3 en 9

• Next lecture: I will answer your questions. Please let me know by email which questions you would like me to answer

November 23, 2010 3

Pure component behavior – pVT diagrams

November 23, 2010 4

Phase behavior of pure substances

sublimation pressure curve

p

T

critical point

triplepoint

vapo

r pressu

recu

rvemel

ting

curv

e

p

v

critical point

triple point

TcTtr

solid liquid

vaporsolid

liquid

vapor

pc

ptr

2 phase LV

2 phase SLLS

isotherm

November 23, 2010 5

Examples of equations of state•Ideal gas equation of state

•Virial equation of state (for non-ideal gasses)

•Van der Waals equation of state (for non-ideal gasses)

2 31 ( ) ( ) ( ) ........pVZ B T p C T p D T pnRT

V 1pZnRT

2

2

n ap V nb nRTV

November 23, 2010 6

The Van der Waals equation of state

The first EOS able to describe vapor and liquids was proposed 1873 by Johannes Diderik van der Waals (Nobel price 1910)

2

RT apv b v

(a 2-parameter EOS: a,b)

the volume accessible to molecules is reduced by b, the volume occupied by molecules

pressure of a real fluid is reduced by the attraction of molecules among each other

free volume =

volume occupied per mole = b

v b

pressure attraction

• Parameter a and b can be determined by fitting the EOS to exp. data, or, more commonly, from critical point data (Tc , pc )

• The VdW EOS self is not accurate and is merely of historical interest

November 23, 2010 7

Van der Waals isotherms

p

v

2

RT apv b v

November 23, 2010 8

Stable, metastable and instable regionsp

v

T=const. (isotherm)

instable region for this

isotherm

metastable region for this

isotherm

stable one-phase region for this

isotherm

stableone-phase

0T

pv

stable two-phase points (liquid and vapor) for this isotherm

0T

pv

Van der Waals isotherm

2

RT apv b v

November 23, 2010 9

The principle of corresponding statesVan der Waals parameter a and b can be determined from critical point data (Tc , pc )

Starting from the conditions for the critical point

2

2

0

0

T

T

pv

pv

2 3

2

32 4

20

2 60

T cc

cT c

p RT av vv b

p RT av vv b

2

RT apv b v

with

2 2276418

c

c

c

c

R Tap

RTbp

The 2 parameters a and b are then defined in terms of (Tc , pc )

2''

64/278/1

RR

RR

vvTp

where reduced variables are defined ascR TTT

'R c cv p v RT

cR ppp

A generalized from of the VdW EOS can then be formulated

November 23, 2010 10

The principle of corresponding statesThe corresponding states principle can visualized in diagrams given in terms of the reduced variables

line is a guide- to-the-eye, not the VdW EOS


Redlich-Kwong EOS• The Redlich-Kwong EOS is an empirical modification of the VdW EOS• It improves the description of real components• It is also a 2-parameter EOS (with parameters a and b)

• a and b of real substances follow from the critical point data (Tc , pc )

RT ap

v b v v b T

2 5 2

0.42748 c

c

R Tap

0.08664 c

c

RTbp


The principle of corresponding states


Corresponding states


Changes of S and H with T and p

,

,

,

,

p N

T N

p N

T N

ST

Sp

HT

Hp


Changes of S and H with T and p

,

,,

,

,,

p

p N

p NT N

pp N

p NT N

CST T

S Vp T

H CT

H VV Tp T

De afleiding van de onderste formule

zie de laatste slide


Enthalpy departure

,,

,

,0

voor een ideaal gas: 0

dus: ( , ) ( )

( ) is de enthalpie die het systeem zou hebben als het een ideaal gas zou zijn

p NT N

T N

pIG

p N

IG

h vv Tp T

hp

vh T p h T v T dpT

h T

dus ( ) hangt dus NIET van de druk af !!!IGh T


Corresponding state principle for departure functions

Combining the principle of corresponding states with the concept of departure functions gives dimensionless charts, that are easy to use for specified T and p

See table A-4 (Moran & Shapiro) for as a function of (pR , TR )

See table A-5 (Moran & Shapiro) for as a function of (pR , TR )

c

ig

TRhh

Rss ig

cR T

TT

cR p

pp


Enthalpy departure in corresponding states

c

ig

TRhh

caution

is equal to minus the values from this table

asterisk * (just as ig) is for ideal gas


Entropy-departure in corresponding states


Exercise•Stikstof komt een turbine binnen bij 100bar en 300K en komt de turbine uit bij 40bar en 245K. Er vind geen warmteuitwisseling plaats met de omgeving. Gegeven is dat Tc =126K en pc =33.9 bar. De cp van stikstof is 28 J mol-1 K-1. Bereken de geleverde arbeid in kJ/kmol en de entropieproductie in kJ/(kmol K) met behulp van de departure grafieken.

e

eeei

iiicvcv

cv gzhmgzhmWQdt

dE2

V2

V 22

met 0cv cvi i e e cv cv

trans

dS Q m s m sdt T


2 2

1 1

1 2 2 1

1 2,

1 1 2 1

en

28*(300 245) J/mol 1540 J/mol

300 100 245 402.38 2.95 1.94 1.18126 33.9 126 33.9

met behulp van de tabell

CV CV

T TIG IG

pp NT T

R R R R

W h h s sm m

hh h dT c dTT

T p T p

2

1

1 2

1 2

IG2 1

,

en: 0.5 0.31

en 0.21 0.14

J J Jdus 1540 8.31 126K 0.5 0.31 1341 mol mol K mol

s

IG IG

c c

IG IG

CV

TIG

p NT

h h h hRT RT

s s s sR R

Wm

S Ss dTT p

2 2 2

1 1 1

2 2

1 1, ,

245 40 J Jln ln 28ln 8.31ln 1.94300 100 mol K mol K

J J Jdus 1.94 8.31 (0.14 0.21) 2.5mol K mol K mol K

p T pp

pp T pT N p N

CV

c T pVdp dT dp c RT T T p

m


2 2

1 1

2 2

1 1

2 2 2 1 1 1, ,

,

2

1

Voor een ideaal gas:

( , ) ( , )

ln ln

T pIG IG

p NT p T N

T pp

T p p N

p

S Ss T p s T p dT dpT p

c VdT dpT T

Tc RT

2

1

pp

Voor elk ideaal gas

Alleen voor ideaal gas met cp onafhankelijk van de temperatuur


Verandering van H met p

, , ,

,,

,,

T N T N T N

p NT N

p NT N

G U TS pV H TS

G H STp p p

H VV Tp T

H VV Tp T

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